Mathematical modeling of kiwi fruits vacuum drying

Harvesting of Kiwifruit (Actinidiadeliciosa, family: Actinidiaceae) is usually performed in mid-October in Iran. The average weight of this fruit is about 70 g. Hayward is the most popular kiwifruit variety in the world mainly due to its large size, oval shape and high shelf life. Drying fresh products is a long-standing method for conservation of food products. This method reduces water-borne and microbiological activities in fresh products while only minor physical and chemical changes occur in these products. Drying, therefore, is regarded as a common method used for food product conversation. There have been several researches on modeling drying of food products. Wang et al. (2007) worked on a mathematical modeling for drying apple slices in a hot air drying process and determining the effective thermal diffusivity. These researchers stated that Midili model was found to be the best for predicting the moisture content changes during drying. Torgul (2005) confirmed this finding in modeling the drying of apple slices in an infrared drying system. How ever not much research has been carned out on drying kiwifruit slices. Therefore, in this research, the drying process of kiwi slices in a vacuum dryer was examined in order to understand their behavior during the process and to determine a best predictive model for drying and also study the diffusivity coefficient for this product.

In thes research Hayward variety of kiwifruit for was used Sinco this variety is commonly grown in Iran. The fruits were purchased from local market in mid-October and transferred to a cool storage (50 C) in a lab at the Department of Biosystems Engineering at the Ferdowsi University of Mashhad. The samples used in this study were of medium size and suitable for cutting in a cylinder-shape cutter. The initial moisture content was determined by so-called oven-drying method on wet basis according to the following equation (Mohesnin, 1986):〖%MC〗_wb=(Initial weight-Final weight (after drying in oven))/(Initial weight) 100 (1) The moisture content was determined as 80.23% on a wet basis. The kiwifruits were sliced at 3 mm thickness using a 35 mm-diameter cylinder and weighed with a digital scale. The slices were moved out of the dryer and weighed every 30 min to monitor their moisture content. Weighing continued until the sample’s moisture content reached to 15-20% on a wet basis. Moisture ratio of kiwifruit slices during drying process was determined according to the follow equation:……………… ……….(2) where MR is moisture ratio (dimensionless), Mt is moisture content at any desirable time, Me equilibrium moisture content, percent, dry basis, and M0 is the initial moisture content (kg H2O/kg of dry matter). The value of Me is very small compared with Mt and M0, hence, the error involved in the simplification of above equation by omitting Me is negligible. The experimental drying data were fitted in various drying models commonly used for monitoring the trend of being-dried products. A few of which models are as follows: MR=exp (-kt): Newton model MR = exp (_ktn): Page model MR = 1 + a.t + bt2: Wang and Singh model MR = a.exp (_ktn) + b.t: Midilli model In this research, two statistical proameters were used to evaluate the goodness of fit of the tested models to the experimental data: the coefficient of determination (R2) and root mean square error (RMSE) between the experimental and the predicted moisture ratio values. Diffusivity coefficient for each slice was determined from the following equation: MR=8/π^2 ∑_(n=0)^∞▒〖1/(2n+1)^2 exp⁡[-(π^2 (2n+1)^2)/4 (D_eff t)/a^2] 〗 (3) where a is sample thickness (in meter), t drying time (in seconds), n is the number of observations and Deff is effective diffusivity coefficient (in m2.s-1). In long drying process, the following simplified equation is used: MR=8/π^2 exp⁡[-(〖π^2 D〗_eff t)/(4a^2)] (4) The diffusivity coefficient is the slope of the straight line when experimental drying data in terms of Ln (MR) is plotted versus drying time (t).

The results of this research revealed that the best prediction curve of moisture content against time was drawn using of MTLAB software. In this regards the rational function with first degree in both numerator and denominator and the third degree polynomial function with maximum coefficient of determination (R2) of 0.9991 and 0.9977 and minimum root mean square error (RMSE) of 0.01267 and 0.02412 were the best prediction models, respectively (Table 2). Furthermore, the drying time becomes shorter as the thickness of kiwifruit slices becomes thinner. This is mainly due to the higher thermal gradient within the thinner slices and hence faster moisture removal due evaporation. The heat diffusivity coefficient was also determined from “Ln (MR) – Time” curves (Figure 3). It was observed that with increase of fruit’s thickness, the heat diffusivity coefficient increases. This phenomenon may be related to the molecular dynamics and the surface tension of materials being dried. In other words the minimum and maximum values of the diffusivity coefficient were observed as 2.0904E-6 and 7.1303E-6 m2.s-1 for fruit thicknesses of 3 and 9 mm, respectively (table 3).

The trend of moisture content evolution against drying time during vacuum drying of kiwifruit was investigated using MTLAB software. Different prediction models were examined for the prediction of moisture removal during vacuum drying of kiwifruit. The rational and polynomial functions were determined as the most accurate prediction models with the coefficient of determination (R2) of higher than 0.99 and RMSE of about 0.02. Furthermore, the heat diffusivity coefficient of kiwifruit slices was investigated as a function of slice thickness. A general increasing trend observed for this coefficient as the thickness of the slices increased.